Finite-dimensional representations of the integers Two questions.


*

*How do I find all finite-dimensional linear representations of $\mathbb{Z}$, the additive group of the integers? I know that all finite-dimensional differentiable representations of the reals under addition are given by sending $t$ to $\exp(tA)$ where $A$ is a linear operator on the vector space of given dimension. So I can restrict those representations to the subgroup $\mathbb{Z}$, but are there more?

*What about the integers $\mod m$ under addition? How do I find all their finite-dimensional linear representations?
 A: Figuring this out for $\Bbb Z$ is easy because the group $\Bbb Z$ is generated by the single element $1$.  In particular, for any representation $\rho:\Bbb Z \to \Bbb C^{n \times n}$, the following holds: 


*

*$\rho(1)$ is an invertible matrix

*for any $k \in \Bbb Z$, we have
$$
\rho(k) = \rho(k \cdot 1) = \rho(1)^k
$$


Thus, every representation of $\Bbb Z$ can be written in the form
$$
\rho(k) = A^k
$$
where $A$ is an invertible matrix.  Moreover, if we can find a matrix $B$ for which $\exp(B) = A$ (which we can, as long as $A$ is invertible), then this becomes
$$
\rho(k) = \exp(kB)
$$
so indeed every representation has the form you mentioned.

$\Bbb Z_m$ also is generated by $1$, but it has the additional property that $m \cdot 1 = 0$.  Thus, the representations of $\Bbb Z_m$ are precisely those of the form
$$
\rho(k) = A^k
$$
for which $A^m = I$ ($I$ is the identity matrix).  If $C$ is one of the "logarithms of the identity" (i.e. $\exp(C) = I$), then we can get the representation
$$
\rho(k) = [\exp(C/m)]^k = \exp[(k/m)C]
$$ 
